The House of Bishops will consider on Monday whether gay clergy should still be expected to remain celibate but should no longer be quizzed about their personal lives.
Labour MP Ben Bradshaw, who belongs to parliament's ecclesiastical committee and is in a civil partnership, was quoted by The Times as saying such a move would be "progress".
He said: "It is progress for them to stop asking the celibacy question but it still leaves the Church of England policy based essentially on dishonesty and encouraging its clergy to lie.
"There is a growing sense that if the Church can't sort this out for themselves, then parliament may have to do it for them."
Under current rules, gay and lesbian clergy are requested to promise to be celibate when they apply for ordaination or seek promotion to positions such as a bishop.
The Church of England has been debating for three years how to respond to same-sex marriage and the House of Bishops' recommendations will be considered by the General Synod next month.
Clergy in the Church are currently forbidden from conducting same-sex marriage services or marrying their same-sex partners.
Chief executive of the lobby group Christian Concern, Andrea Williams, told The Times the Church should keep to the view marriage is "the only place of sexual expression ...between one man and one woman in a lifelong union.
The Church of England told the newspaper: "The House of Bishops meets tomorrow and until [then] any claims as to decisions are pure speculation."